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Free Resolutions of Parameter Ideals for Some Rings with Finite Local Cohomology

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Free Resolutions of Parameter Ideals for Some Rings with Finite Local Cohomology FREE RESOLUTIONS OF PARAMETER IDEALS FOR SOME RINGS WITH FINITE LOCAL COHOMOLOGY HAMID RAHMATI Abstract. Let R be a d-dimensional local ring, with maximal ideal m, con- taining a field and let x1; : : : ; xd be a system of parameters for R. If depth R ≥ d−1 d − 1 and the local cohomology module Hm (R) is finitely generated, then i i there exists an integer n such that the modules R=(x1; : : : ; xd) have the same Betti numbers, for all i ≥ n. Introduction Throughout this note R is a local noetherian ring with maximal ideal m. Let M be a finitely generated R-module and let F be a minimal free resolution of R P1 i M. The Poincar´eseries of M is the formal power series PM (t) = i=0(rank Fi)t . i F We let ΩR(M) denote the ith syzygy of M, that is to say Coker @i+1. Let x = n x1; : : : ; xd be a system of parameters for R. For each n 2 N, let x denote the n n sequence x1 ; : : : ; xd . Several classical results in local algebra establish that ideals contained in the large powers of the maximal ideal exhibit a similar behavior. In his thesis, [7], Y. H. Lai considers the following question, which he attributes to D. Katz: Do the Poincar´eseries of the modules R=(xn) behave uniformly, for n large enough? Clearly, the answer is yes if R is Cohen-Macaulay: In this case xn is a regular sequence, for each n ≥ 1, so R=(xn) is resolved by a Koszul complex on d elements. It is also not difficult to obtain a positive answer when dim R = 1. Lai proves that if dim R = 2, depth R = 1 and the local cohomology module 1 Hm(R) is finitely generated, then for n large enough one has R 2 2 R P n (t) = 1 + 2t + t + t P 1 (t): R=(x ) Hm(R) Now we state our main result. It evidently covers the first two cases mentioned above. Also part (ii) of our main theorem generalizes Lai's result, since the Canon- ical Element Conjecture holds for 2-dimensional rings. Main Theorem. Let R be d-dimensional local ring with maximal ideal m. If d−1 d − depth R ≤ 1 and the R-module H = Hm (R) is finitely generated, then there exists an integer n, such that for each system of parameters x for R contained in mn the following assertions hold d+1 ∼ d−1 (i) ΩR (R=(x)) = ΩR (H). (ii) If in addition, the Canonical Element Conjecture holds for R, then one has R d 2 R PR=(x)(t) = (1 + t) + t PH (t): 1991 Mathematics Subject Classification. Primary 13D02, 13D40; Secondary 13H10. 1 2 HAMID RAHMATI The Cohen-Macaulay defect of R is the number cmd R = dim R − depth R. One always has cmd R ≥ 0, and equality characterizes Cohen-Macaulay rings. On the i other hand, Cohen-Macaulay rings are also characterized by the equality Hm(R) = 0 for all i 6= d. Comparing these conditions with the hypotheses on R in the theorem, one may say that these hypotheses, in some sense, give the least possible extension of Cohen-Macaulay rings. 1. Free resolutions of almost complete intersection ideals Let y = y1; : : : ; yn be a sequence in R. We let K(y; M) denote the Koszul complex on y with coefficients in M. Set Hi(y; M) = Hi(K(y; M)): P1 i P1 i If P (t) = i=0 ait and Q(t) = i=0 bit are formal power, we write P 4 Q to indicate that ai ≤ bi holds for all i ≥ 0. The main result of this section is the following theorem: Theorem 1.1. Let R be a d-dimensional ring and let x be a system of parameters for R. Set Hi = Hi(x; R) for i = 1; : : : ; d. One then has cmd R X PR (t) (1 + t)d + ti+1 PR (t): R=(x) 4 Hi i=1 Moreover, if cmd R ≤ 1 and the Canonical Element Conjecture holds for R, then one has PR (t) = (1 + t)d + t2 PR (t): R=(x) H1 Under the hypotheses of the second part of the theorem, x generates an almost complete intersection ideal. Some of the discussion below is carried out in this more general framework. Recall that an ideal I is called almost complete intersection if grade I ≥ µ(I) − 1, where grade I is the maximal length of an R-regular sequence in I and µ(I) is the number of minimal generators of I. Now we give an example that shows the inequality in Theorem 1.1 can be strict if depth R < d − 1. Example 1.2. Set R = k[[a; b; c]]=(ac; bc; c2), then dim R = 2 and depth R = 0. Consider the system of parameters x = a; b. Using Macaulay 2 we get: P R (t) = 1 + 3t + 6t2 + 13t3 + 28t4 + ··· H2 P R (t) = 3 + 7t + 12t2 + 26t3 + 56t4 + ··· H1 R 2 3 4 PR=(x)(t) = 1 + 2t + 3t + 7t + 15t + ··· : Thus one has R 2 3 4 PR=(x)(t) ≺ 1 + 2t + 4t + 8t + 15t + ··· = (1 + t)2 + t2P R (t) + t3P R (t): H1 H2 Let X be a complex of R-modules and let @X denote its differential, set sup H(X) = supfn 2 Z j Hn(X) 6= 0g: Let Σt denote the shift functor defined by t ΣX t X (Σ X)n = Xn−t and @n = (−1) @n−t: FREE RESOLUTIONS OF PARAMETER IDEALS 3 Let α: X ! Y be a morphism of complexes. Recall that mapping cone of α is defined to be the complex C such that Cn = Xn−1 ⊕ Yn and C X Y @n ((x; y)) = (−@n−1(x);@n (y) + αn−1(x)) for all (x; y) 2 Cn: A quasi-isomorphism is a morphism of complexes that induces isomorphism in homology in all degrees. Lemma 1.3. Let X be a complex with s = sup H(X) < 1 and let F be a free s resolution of Hs(X). There exists a morphism of complexes α:Σ F ! X such that the mapping cone X0 of α satisfies ( 0 ∼ 0 i ≥ s Hi(X ) = Hi(X) i ≤ s − 1: Proof. Let τ≥s(X) be the complex X · · · −! Xs+2 −! Xs+1 −! ker @s −! 0 and ι: τ≥s(X) ! X be the inclusion map. Since F is a bounded below complex of free modules, there is a morphism of complexes β : F ! τ≥s(X) such that the following diagram is commutative β s ι Σ F / τ≥s(X) / X JJ JJ JJ π s JJ Σ JJ% s Σ Hs(X); where : F ! Hs(X) and π are quasiisomorphism. Set α = ιβ and let C be the mapping cone of α. One has an exact sequence 0 −! X −! X0 −! Σs+1F −! 0 of complexes. It induces an exact sequence of homology modules 0 s+1 Hi(α) ···! Hi+1(X) ! Hi+1(X ) ! Hi+1(Σ F ) −−−−! Hi(X) !··· : From the construction of α one sees that Hs(α) is an isomorphism, and since s+1 ∼ Hi(Σ F ) = Hi−s−1(F ) = 0 for all i 6= s + 1, we get the desired result. Remark 1.4. Assume X is a complex of free modules such that Xi = 0, for i < 0, i and Hi(X) = 0, for i > s, where s is some positive integer. Let F be a free resolution of Hi(X), for i = 1; : : : ; s. Applying Lemma 1.3 s times, one gets a free 1 s resolution G of H0(X) such that Gi = Xi ⊕ Fi−2 ⊕ · · · ⊕ Fi−s−1. However, even if the complexes X; F 1;:::;F s are minimal, G need not be minimal; see Example 1.2. Now we show the relation between the Poincar´eseries of an almost complete intersection ideal and the Poincar´eseries of its first Koszul homology module. Lemma 1.5. Let I be an almost intersection ideal of R and let y = y1; : : : ; yr be a minimal set generators for I. Set H = H1(y; R). One then has r+1 r−1 ΩR (R=I) = ΩR (H) R r+1 R PR=I (t) = Q(t) + t PN (t); r−1 where Q(t) is a polynomial of degree r and N = ΩR (H). 4 HAMID RAHMATI Proof. Let K denote the Koszul complex of y with coefficients in R. Since grade I ≥ r − 1, one has Hi(y; R) = 0 for i 6= 0; 1. Let F be a minimal free resolution of H. Let α:ΣF ! K be the map from Lemma 1.3 and let C be the mapping cone of α. The complex C is a complex of free modules and has only one nonvanishing ∼ ∼ homology module namely, H0(C) = H0(y; R) = R=I, so C is a free resolution of R=I. By construction of mapping cones the complex C is minimal if and only if αF ⊆ mK. Since F is minimal and Cm = Fm for all m ≥ r + 1, non-minimality can only happen in the first r + 1 degrees, therefore r+1 r ΩR (R=I) = ΩR(H): This completes the proof of the first equality. The second equality follows from the first one. 1.6. For a system of parameters x for R, let Fx be a free resolution of R=(x) and let γx : Kx = K(x; R) ! Fx be a lifting of the map Kx ! R=(x).
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